Precalculus: a prelude to calculus 1st edition paper

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cos Θ, sin Θ

slope

tan Θ

Θ
1

Precalculus
A Prelude to Calculus
with

Student Solutions Manual

Sheldon Axler
San Francisco State University

JOHN WILEY & SONS, INC.


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About the Cover

The diagram on the cover contains the crucial definitions of trigonometry.
The 1 shows that the trigonometric functions are defined in the context of
the unit circle. The arrow shows that angles are measured counterclockwise
from the positive horizontal axis. The point labeled (cos θ, sin θ) shows that
cos θ is the first coordinate of the endpoint of the radius corresponding to
the angle θ, and sin θ is the second coordinate of this endpoint. Because this
endpoint is on the unit circle, the identity cos2 θ + sin2 θ = 1 immediately
follows. The equation slope = tan θ shows that tan θ is the slope of the
sin θ
radius corresponding to the angle θ; thus tan θ = cos θ .

This book is printed on acid free paper. ∞
Copyright © 2009 John Wiley & Sons, Inc. All rights reserved. No part of this
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ISBN-13
ISBN-13
ISBN-13

978-0470-41674-7 (hardcover)
978-0470-18072-3 (softcover)
978-0470-41813-0 (binder ready)

Printed in the United States of America
10 9 8 7 6 5 4 3 2 1


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About the Author

Sheldon Axler is Dean of
the College of Science & Engineering at San Francisco
State University, where he
joined the faculty as Chair of the Mathematics Department in 1997.
Axler was valedictorian of his high school in Miami, Florida. He received
his AB from Princeton University with highest honors, followed by a PhD in

Mathematics from the University of California at Berkeley.
As a postdoctoral Moore Instructor at MIT, Axler received a university-wide
teaching award. Axler was then an assistant professor, associate professor,
and professor at Michigan State University, where he received the first J.
Sutherland Frame Teaching Award and the Distinguished Faculty Award.
Axler received the Lester R. Ford Award for expository writing from the
Mathematical Association of America in 1996. In addition to publishing numerous research papers, Axler is the author of Linear Algebra Done Right
(which has been adopted as a textbook at over 225 universities and colleges) and co-author of Harmonic Function Theory (a graduate/research-level
book).
Axler has served as Editor-in-Chief of the Mathematical Intelligencer and
as Associate Editor of the American Mathematical Monthly. He has been a
member of the Council of the American Mathematical Society and a member
of the Board of Trustees of the Mathematical Sciences Research Institute.
Axler currently serves on the editorial board of Springer’s series Undergraduate Texts in Mathematics, Graduate Texts in Mathematics, and Universitext.

v


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Contents

About the Author

v

Preface to the Instructor xv
Acknowledgments

xx

Preface to the Student
0 The Real Numbers

xxiii

1

0.1 The Real Line 2
Construction of the Real Line 2
Is Every Real Number Rational? 3
Problems 6

0.2 Algebra of the Real Numbers 7
Commutativity and Associativity 7
The Order of Algebraic Operations 8
The Distributive Property 10
Additive Inverses and Subtraction 11
Multiplicative Inverses and Division 12
Exercises, Problems, and Worked-out Solutions 14

0.3 Inequalities 18
Positive and Negative Numbers 18
Lesser and Greater 19
Intervals 21
Absolute Value 24
Exercises, Problems, and Worked-out Solutions 26

Chapter Summary and Chapter Review Questions 32

vi


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Contents

1 Functions and Their Graphs 33
1.1 Functions

34

Definition and Examples 34
Equality of Functions 36
The Domain of a Function 37
Functions via Tables 38
The Range of a Function 38
Exercises, Problems, and Worked-out Solutions 41

1.2 The Coordinate Plane and Graphs 47
The Coordinate Plane 47
The Graph of a Function 49
Determining a Function from Its Graph 51
Which Sets Are Graphs of Functions? 53
Determining the Range of a Function from Its Graph 53
Exercises, Problems, and Worked-out Solutions 54

1.3 Function Transformations and Graphs 62
Shifting a Graph Up or Down 62
Shifting a Graph Right or Left 63

Stretching a Graph Vertically or Horizontally 65
Reflecting a Graph in the Horizontal or Vertical Axis 67
Even and Odd Functions 69
Exercises, Problems, and Worked-out Solutions 71

1.4 Composition of Functions 81
Definition of Composition 81
Order Matters in Composition 82
Decomposing Functions 83
Composing More than Two Functions 84
Exercises, Problems, and Worked-out Solutions 85

1.5 Inverse Functions 90
The Inverse Problem 90
One-to-one Functions 91
The Definition of an Inverse Function 92
The Domain and Range of an Inverse Function 94
The Composition of a Function and Its Inverse 95
Comments about Notation 97
Exercises, Problems, and Worked-out Solutions 98

vii


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viii

Contents

1.6 A Graphical Approach to Inverse Functions 104
The Graph of an Inverse Function 104
Inverse Functions via Tables 106
Graphical Interpretation of One-to-One 106
Increasing and Decreasing Functions 108
Exercises, Problems, and Worked-out Solutions 110

Chapter Summary and Chapter Review Questions 115
2 Linear, Quadratic, Polynomial, and Rational Functions 117
2.1 Linear Functions and Lines 118
Slope 118
The Equation of a Line 119
Parallel Lines 122
Perpendicular Lines 124
Exercises, Problems, and Worked-out Solutions 127

2.2 Quadratic Functions and Parabolas 134
The Vertex of a Parabola 134
Completing the Square 136
The Quadratic Formula 139
Exercises, Problems, and Worked-out Solutions 141

2.3 Integer Exponents 147
Exponentiation by Positive Integers 147
Properties of Exponentiation 148
Defining x 0

149

Exponentiation by Negative Integers 150

Manipulations with Powers 151
Exercises, Problems, and Worked-out Solutions 153

2.4 Polynomials 159
The Degree of a Polynomial 159
The Algebra of Polynomials 160
Zeros and Factorization of Polynomials 162
The Behavior of a Polynomial Near ±∞

165

Graphs of Polynomials 168
Exercises, Problems, and Worked-out Solutions 170


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Contents

2.5 Rational Functions 175
Ratios of Polynomials 175
The Algebra of Rational Functions 176
Division of Polynomials 177
The Behavior of a Rational Function Near ±∞

180

Graphs of Rational Functions 183
Exercises, Problems, and Worked-out Solutions 184

2.6 Complex Numbers∗

190

The Complex Number System 190
Arithmetic with Complex Numbers 191
Complex Conjugates and Division of Complex Numbers 192
Zeros and Factorization of Polynomials, Revisited 195
Exercises, Problems, and Worked-out Solutions 198

2.7 Systems of Equations and Matrices∗

203

Solving a System of Equations 203
Systems of Linear Equations 205
Matrices and Linear Equations 209
Exercises, Problems, and Worked-out Solutions 216

Chapter Summary and Chapter Review Questions 222
3 Exponents and Logarithms 224
3.1 Rational and Real Exponents 225
Roots 225
Rational Exponents 228
Real Exponents 230
Exercises, Problems, and Worked-out Solutions 232

3.2 Logarithms as Inverses of Exponentiation 238
Logarithms Base 2 238
Logarithms with Arbitrary Base 239

Change of Base 241
Exercises, Problems, and Worked-out Solutions 243

3.3 Algebraic Properties of Logarithms 248
Logarithm of a Product 248
Logarithm of a Quotient 249
* Section

can be skipped if focusing only on material needed for first-semester calculus.

ix


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x

Contents

Common Logarithms and the Number of Digits 250
Logarithm of a Power 251
Exercises, Problems, and Worked-out Solutions 252

3.4 Exponential Growth 259
Functions with Exponential Growth 260
Population Growth 262
Compound Interest 264
Exercises, Problems, and Worked-out Solutions 269

3.5 Additional Applications of Exponents and Logarithms 275

Radioactive Decay and Half-Life 275
Earthquakes and the Richter Scale 277
Sound Intensity and Decibels 279
Star Brightness and Apparent Magnitude 280
Exercises, Problems, and Worked-out Solutions 282

Chapter Summary and Chapter Review Questions 288
4 Area, e, and the Natural Logarithm 290
4.1 Distance, Length, and Circles 291
Distance between Two Points 291
Midpoints 292
Distance between a Point and a Line 293
Circles 294
Length 296
Exercises, Problems, and Worked-out Solutions 298

4.2 Areas of Simple Regions 304
Squares 304
Rectangles 305
Parallelograms 305
Triangles 305
Trapezoids 306
Stretching 307
Circles 308
Ellipses 310
Exercises, Problems, and Worked-out Solutions 313

4.3 e and the Natural Logarithm 321
Estimating Area Using Rectangles 321

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Contents

Defining e

323

Defining the Natural Logarithm 326
Properties of the Exponential Function and ln 327
Exercises, Problems, and Worked-out Solutions 329

4.4 Approximations with e and ln

336

Approximation of the Natural Logarithm 336
Inequalities with the Natural Logarithm 337
Approximations with the Exponential Function 338
An Area Formula 339
Exercises, Problems, and Worked-out Solutions 342

4.5 Exponential Growth Revisited 346
Continuously Compounded Interest 346
Continuous Growth Rates 347
Doubling Your Money 348
Exercises, Problems, and Worked-out Solutions 350

Chapter Summary and Chapter Review Questions 355

5 Trigonometric Functions 357
5.1 The Unit Circle 358
The Equation of the Unit Circle 358
Angles in the Unit Circle 359
Negative Angles 361
Angles Greater Than 360◦

362

Length of a Circular Arc 363
Special Points on the Unit Circle 364
Exercises, Problems, and Worked-out Solutions 365

5.2 Radians

371

A Natural Unit of Measurement for Angles 371
Negative Angles 374
Angles Greater Than 2π

375

Length of a Circular Arc 376
Area of a Slice 376
Special Points on the Unit Circle 377
Exercises, Problems, and Worked-out Solutions 378

5.3 Cosine and Sine 383
Definition of Cosine and Sine 383

xi


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xii

Contents

Cosine and Sine of Special Angles 385
The Signs of Cosine and Sine 386
The Key Equation Connecting Cosine and Sine 388
The Graphs of Cosine and Sine 389
Exercises, Problems, and Worked-out Solutions 391

5.4 More Trigonometric Functions 396
Definition of Tangent 396
Tangent of Special Angles 397
The Sign of Tangent 398
Connections between Cosine, Sine, and Tangent 399
The Graph of Tangent 399
Three More Trigonometric Functions 401
Exercises, Problems, and Worked-out Solutions 402

5.5 Trigonometry in Right Triangles 408
Trigonometric Functions via Right Triangles 408
Two Sides of a Right Triangle 410
One Side and One Angle of a Right Triangle 411
Exercises, Problems, and Worked-out Solutions 412

5.6 Trigonometric Identities 417
The Relationship Between Cosine and Sine 417
Trigonometric Identities for the Negative of an Angle 419
Trigonometric Identities with

π
2

420

Trigonometric Identities Involving a Multiple of π

422

Exercises, Problems, and Worked-out Solutions 426

5.7 Inverse Trigonometric Functions 431
The Arccosine Function 431
The Arcsine Function 434
The Arctangent Function 436
Exercises, Problems, and Worked-out Solutions 439

5.8 Inverse Trigonometric Identities 442
The Arccosine, Arcsine, and Arctangent of −t:
Graphical Approach 442
The Arccosine, Arcsine, and Arctangent of −t:
Algebraic Approach 444
Arccosine Plus Arcsine 445
The Arctangent of

1
t

445

Composition of Trigonometric Functions and Their Inverses 446


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Contents xiii

More Compositions with Inverse Trigonometric Functions 447
Exercises, Problems, and Worked-out Solutions 450

Chapter Summary and Chapter Review Questions 454
6 Applications of Trigonometry 456
6.1 Using Trigonometry to Compute Area 457
The Area of a Triangle via Trigonometry 457
Ambiguous Angles 458
The Area of a Parallelogram via Trigonometry 460
The Area of a Polygon 461
Exercises, Problems, and Worked-out Solutions 462

6.2 The Law of Sines and the Law of Cosines 468
The Law of Sines 468
Using the Law of Sines 469
The Law of Cosines 471
Using the Law of Cosines 472

When to Use Which Law 474
Exercises, Problems, and Worked-out Solutions 475

6.3 Double-Angle and Half-Angle Formulas 482
The Cosine of 2θ
The Sine of 2θ

482

483

The Tangent of 2θ

484

The Cosine and Sine of
The Tangent of

θ
2

θ
2

485

487

Exercises, Problems, and Worked-out Solutions 488

6.4 Addition and Subtraction Formulas 496
The Cosine of a Sum and Difference 496
The Sine of a Sum and Difference 498
The Tangent of a Sum and Difference 499
Exercises, Problems, and Worked-out Solutions 501

6.5 Transformations of Trigonometric Functions 506
Amplitude 506
Period 508
Phase Shift 511
Exercises, Problems, and Worked-out Solutions 513


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xiv

Contents

6.6 Polar Coordinates∗

522

Defining Polar Coordinates 522
Converting from Polar to Rectangular Coordinates 523
Converting from Rectangular to Polar Coordinates 524
Graphs of Polar Equations 528
Exercises, Problems, and Worked-out Solutions 530

6.7 Vectors and the Complex Plane∗

533

An Algebraic and Geometric Introduction to Vectors 533
The Dot Product 539
The Complex Plane 541
De Moivre’s Theorem 545
Exercises, Problems, and Worked-out Solutions 546

Chapter Summary and Chapter Review Questions 550
7 Sequences, Series, and Limits 552
7.1 Sequences

553

Introduction to Sequences 553
Arithmetic Sequences 555
Geometric Sequences 556
Recursive Sequences 558
Exercises, Problems, and Worked-out Solutions 561

7.2 Series

567

Sums of Sequences 567
Arithmetic Series 567
Geometric Series 569
Summation Notation 571
Exercises, Problems, and Worked-out Solutions 572

7.3 Limits 577
Introduction to Limits 577
Infinite Series 581
Decimals as Infinite Series 583
Special Infinite Series 585
Exercises, Problems, and Worked-out Solutions 587

Chapter Summary and Chapter Review Questions 590
Index
* Section

593
can be skipped if focusing only on material needed for first-semester calculus.


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Preface to the Instructor

Goals and Prerequisites
This book seeks to prepare students to succeed in calculus. Thus this book focuses
on topics that students need for calculus, especially first-semester calculus. Many
important subjects that should be known by all educated citizens but that are irrelevant to calculus have been excluded.
Precalculus is a one-semester course at most universities. Nevertheless, typical
precalculus textbooks contain about a thousand pages (not counting a student solutions manual), far more than can be covered in one semester. By emphasizing topics
crucial to success in calculus, this book has a more manageable size even though it
includes a student solutions manual. A thinner textbook should indicate to students
that they are truly expected to master most of the content of the book.
The prerequisite for this course is the usual course in intermediate algebra. Many

students in precalculus classes have had a trigonometry course previously, but this
book does not assume that the students remember any trigonometry. In fact the
book is fairly self-contained, starting with a review of the real numbers in Chapter 0,
whose numbering is intended to indicate that many instructors will prefer to skip
this beginning material or cover it quickly.

Inverse Functions
The unifying concept of inverse functions are introduced early in the book in Section 1.5. This crucial idea has its first major use in this book in the definition of
y 1/m as the number that when raised to the mth power gives y (in other words, the
function y → y 1/m is the inverse of the function x → x m ; see Section 3.1). The second major use of inverse functions occurs in the definition of logb y as the number
such that b raised to this number gives y (in other words, the function y → logb y
is the inverse of the function x → bx ; see Section 3.2).
Thus students should be comfortable with using inverse functions by the time
they reach the inverse trigonometric functions (arccosine, arcsine, and arctangent)
in Section 5.7. This familiarity with inverse functions should help students deal with
inverse operations (such as antidifferentiation) when they reach calculus.

Algebraic Properties of Logarithms
Logarithms play a key role in calculus, but many calculus instructors complain that
too many students lack appropriate algebraic manipulation skills with logarithms.
In Chapter 3 logarithms are defined as the inverse functions of exponentiation. The
base for logarithms here is arbitrary, although most of the examples and motivation in Chapter 3 use logarithms base 2 or logarithms base 10. In Chapter 3, the

xv

Chapter 0 could have
been titled
A Prelude to A
Prelude to Calculus.

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xvi

Preface to the Instructor

The initial separation
of logarithms and e
should help students
master both concepts.

algebraic properties of logarithms are easily derived from the algebraic properties
of exponentiation.
The crucial concepts of e and natural logarithms are saved for a later chapter.
Thus students can concentrate in Chapter 3 on understanding logarithms (arbitrary
base) and their properties without at the same time worrying about grasping concepts related to e. Similarly, when natural logarithms arise naturally in Chapter 4,
students should be able to concentrate on issues surrounding e without at the same
time learning properties of logarithms.

Half-life and Exponential Growth
All precalculus textbooks present radioactive decay as an example of exponential
decay. Amazingly, the typical precalculus textbook states that if a radioactive isotope has a half-life of h, then the amount left at time t will equal e−kt times the
ln 2
amount present at time 0, where k = h .
A much clearer formulation would state, as this textbook does, that the amount
left at time t will equal 2−t/h times the amount present at time 0. The unnecessary
use of e and ln 2 in this context may suggest to students that e and natural logarithms
have only contrived and artificial uses, which is not the message that students should
receive from their textbook. Using 2−t/h helps students understand the concept of

half-life, with a formula connected to the meaning of the concept.
Similarly, many precalculus textbooks consider, for example, a colony of bacteria
doubling in size every 3 hours, with the textbook then producing the formula e(t ln 2)/3
for the growth factor after t hours. The simpler and natural formula 2t/3 seems not
to be mentioned in such books. This book presents the more natural approach to
such issues of exponential growth and decay.

Area
About half of calculus (namely, integration) deals with area, but most precalculus
textbooks barely mention the subject.
Chapter 4 in this book builds the intuitive notion of area starting with squares,
and then quickly derives formulas for the area of rectangles, triangles, parallelograms, and trapezoids. A discussion of the effects of stretching either horizontally
or vertically easily leads to the familiar formula for the area enclosed by a circle.
Chapter 4 uses the same ideas to derive the formula for the area inside an ellipse.
Chapter 4 then turns to the question of estimating the area under parts of the
1
curve y = x by using rectangles. This easy nontechnical introduction, with its
emphasis on ideas without the clutter of the notation of Riemann sums, will serve
students well when they reach integral calculus in a later course.

e, The Exponential Function, and the Natural Logarithm
Most precalculus textbooks either present no motivation for e or motivate e via continuously compounding interest or through the limit of an indeterminate expression
of the form 1∞ ; these concepts are difficult for students at this level to understand.
Chapter 4 presents a clean and well-motivated approach to e and the natural
logarithm. We do this by looking at the area (intuitively defined) under the curve
1
y = x , above the x-axis, and between the lines x = 1 and x = c.


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Preface to the Instructor xvii
A similar approach to e and the natural logarithm is common in calculus courses.
However, this approach is not usually adopted in precalculus textbooks. Using obvious properties of area, the simple presentation given here shows how these ideas
can come through clearly without the technicalities of calculus or Riemann sums.
Indeed, this precalculus approach to the exponential function and the natural logarithm shows that a good understanding of these subjects need not wait until the
calculus course. Students who have seen the approach given here should be well
prepared to deal with these concepts in their calculus courses.
The approach taken here also has the advantage that it easily leads, as we will see
in Chapter 4, to the approximation ln(1 + h) ≈ h for small values of h. Furthermore,
r x
≈ er for large values
the same methods show that if r is any number, then 1 + x
of x. A final bonus of this approach is that the connection between continuously
compounding interest and e becomes a nice corollary of natural considerations concerning area.

Trigonometry
Should the trigonometric functions be introduced via the unit circle or via right triangles? Calculus requires the unit-circle approach (because, for example, discussing
the Taylor series for cos x requires us to consider negative values of x and values of
π
x that are more than 2 radians). Thus this textbook uses the unit-circle approach,
but quickly gives applications to right triangles. The unit-circle approach also allows
for a well-motivated introduction to radian measure.
The trigonometry section of this book concentrates almost exclusively on the
functions cosine, sine, and tangent and their inverse functions, with only cursory
mention of secant, cosecant, and cotangent. These latter three functions, which are
simply the multiplicative inverses of the three key trigonometric functions, add little
content or understanding.

Exercises and Problems

Students learn mathematics by actively working on a wide range of exercises and
problems. Ideally, a student who reads and understands the material in a section of
this book should be able to do the exercises and problems in that section without
further help. However, some of the exercises require application of the ideas in a
context that students may not have seen before, and many students will need help
with these exercises. This help is available from the complete worked-out solutions
to all the odd-numbered exercises that appear at the end of each section.
Because the worked-out solutions were written solely by the author of the textbook, students can expect a consistent approach to the material. Furthermore,
students will save money by not having to purchase a separate student solutions
manual.
The exercises (but not the problems) occur in pairs, so that an odd-numbered
exercise is followed by an even-numbered exercise whose solution uses the same
ideas and techniques. A student stumped by an even-numbered exercise should be
able to tackle it after reading the worked-out solution to the corresponding oddnumbered exercise. This arrangement allows the text to focus more centrally on
explanations of the material and examples of the concepts.
My experience with teaching precalculus is that most students read the student
solutions manual when they are assigned homework, even though they are reluctant

Each exercise has a
unique correct answer, usually a number or a function; each
problem has multiple
correct answers, usually explanations or
examples.

This book contains
what is usually a separate book called the
student solutions
manual.

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xviii

Preface to the Instructor
to read the main text. The integration of the student solutions manual within this
book might encourage students who would otherwise read only the student solutions
manual to drift over and also read the main text. To reinforce this tendency, the
worked-out solutions to the odd-numbered exercises at the end of each section are
typeset with a slightly less appealing style (smaller type, two-column format, and
not right justified) than the main text. The reader-friendly appearance of the main
text might nudge students to spend some time there.
Exercises and problems in this book vary greatly in difficulty and purpose. Some
exercises and problems are designed to hone algebraic manipulation skills; other
exercises and problems are designed to push students to genuine understanding
beyond rote algorithmic calculation.
Some exercises and problems intentionally reinforce material from earlier in the
book. For example, Exercise 27 in Section 5.3 asks students to find the smallest
number x such that sin(ex ) = 0; students will need to understand that they want to
choose x so that ex = π and thus x = ln π . Although such exercises require more
thought than most exercises in the book, they allow students to see crucial concepts
more than once, sometimes in unexpected contexts.

A Book Designed to be Read
Mathematics faculty frequently complain, with justification, that most students in
lower-division mathematics courses do not read the textbook. When doing homework, a typical precalculus student looks only at the relevant section of the textbook
or the student solutions manual for an example similar to the homework problem
at hand. The student reads enough of that example to imitate the procedure, does
the homework problem, and then follows the same process with the next homework
problem. Little understanding may take place.

In contrast, this book is designed to be read by students. The writing style and
layout are meant to induce students to read and understand the material. Explanations are more plentiful than typically found in precalculus books, with examples of
the concepts making the ideas concrete whenever possible.

The Calculator Issue
To aid instructors
in presenting the
kind of course they
want, the symbol
appears with exercises and problems
that require students
to use a calculator.

The issue of whether and how calculators should be used by students has generated
immense controversy.
Some sections of this book have many exercises and problems designed for calculators (for example Section 3.4 on exponential growth and Section 6.2 on the law of
sines and the law of cosines), but some sections deal with material not as amenable to
calculator use. The emphasis throughout the text has been on giving students both
the understanding and the skills they need to succeed in calculus. Thus the book
does not aim for an artificially predetermined percentage of exercises and problems
in each section requiring calculator use.
Some exercises and problems that require a calculator are intentionally designed
to make students realize that by understanding the material, they can overcome the
limitations of calculators. As one example among many, Exercise 41 in Section 3.3
asks students to find the number of digits in the decimal expansion of 74000 . Brute
force with a calculator will not work with this problem because the number involved
has too many digits. However, a few moments’ thought should show students that
they can solve this problem by using logarithms (and their calculators!).

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Preface to the Instructor xix
can be interpreted for some exercises, depending on the
The calculator icon
instructor’s preference, to mean that the solution should be a decimal approximation rather than the exact answer. For example, Exercise 3 in Section 4.5 asks how
much would need to be deposited in a bank account paying 4% interest compounded
continuously so that at the end of 10 years the account would contain $10,000. The
exact answer to this exercise is 10000/e0.4 dollars, but it may be more satisfying to
the student (after obtaining the exact answer) to use a calculator to see that approximately $6,703 needs to be deposited. For such exercises, instructors can decide
whether to ask for exact answers or decimal approximations (the worked-out solutions for the odd-numbered exercises will usually contain both).

What to Cover
Different instructors will want to cover different sections of this book. I usually
cover Chapter 0 (The Real Numbers), even though it should be review, because it
deals with familiar topics in a deeper fashion than students may have previously
seen. I frequently cover Section 2.5 (Rational Functions) only lightly because graphing rational functions, and in particular finding local minima and maxima, is better
done with calculus. Many instructors will prefer to skip Chapter 7 (Sequences, Series,
and Limits), leaving that material to a calculus course. A one-semester precalculus
course will probably not have time to cover the sections denoted with an asterisk (∗ );
those sections can safely be skipped by courses focusing only on material needed
for first-semester calculus.

What’s New as Compared to the Preliminary Edition
Numerous improvements have been made throughout the text based upon suggestions from faculty and students who used the Preliminary Edition. For example, the
introduction to e now gives instructors a gentler path to help lead students to discover this remarkable number. More exercises and problems have been added to
many sections.
Some faculty requested coverage of additional topics because their precalculus
courses serve other purposes beyond preparing students for first-semester calculus.
Thus three new optional sections have been added, dealing with complex numbers,

systems of equations and matrices, and vectors.
A major redesign using full color has led to considerable improvements in the
appearance of the book.
Finally, a comprehensive index now allows users to locate topics within the book
quickly.

Comments Welcome
I seek your help in making this a better book. Please send me your comments and
your suggestions for improvements. Thanks!
Sheldon Axler
San Francisco State University
e-mail:
web page: www.axler.net

Regardless of what
level of calculator use
an instructor expects,
students should not
turn to a calculator to
compute something
like cos 0, because
then cos has become
just a button on the
calculator.


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Acknowledgments

Most of the results in
this book belong to
the common heritage
of mathematics, created over thousands
of years by clever
and curious people.

As usual in a textbook, as opposed to a research article, little attempt has
been made to provide proper credit to the original creators of the ideas presented in this book. Where possible, I have tried to improve on standard
approaches to this material. However, the absence of a reference does not
imply originality on my part. I thank the many mathematicians who have
created and refined our beautiful subject.
I chose Wiley as the publisher of this book because of the company’s commitment to excellence. The people at Wiley have made outstanding contributions to this project, providing wise editorial advice, superb design expertise, high-level production skill, and insightful marketing savvy. I am truly
grateful to the following Wiley folks, all of whom helped make this a better and more successful book than it would have been otherwise: Angela
Battle, Jeff Benson, Melissa Edwards, Jaclyn Elkins, Jessica Jacobs, Madelyn
Lesure, Chelsee Pengal, Laurie Rosatone, Christopher Ruel, Ken Santor, Anne
Scanlan-Rohrer, Elle Wagner.
The accuracy checkers (Victoria Green, Celeste Hernandez, Nancy Matthews, Yan Tian, and Charles Waiveris) and copy editors (Katrina Avery and
Patricia Brecht) excelled at catching my mathematical and linguistic mistakes.
The instructors and students who used the Preliminary Edition of this
book provided wonderfully useful feedback. Numerous reviewers gave me
terrific suggestions as the book progressed through various stages of development. I am grateful to all the class testers and reviewers whose names are
listed on the following pages.
Like most mathematicians, I owe thanks to Donald Knuth, who invented
TEX, and to Leslie Lamport, who invented LATEX, which I used to typeset this
book.
Thanks also to Wolfram Research for producing Mathematica, which is
the software I used to create the graphics in this book.
My awesome partner Carrie Heeter deserves considerable credit for her astute advice and continual encouragement throughout the long book-writing
process.

Many thanks to all of you!

xx


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Acknowledgments xxi

Class Testers and Reviewers
• Alison Ahlgren, University of Illinois,
Urbana-Champaign

• Thomas English, Penn State University, Erie

• Margo Alexander, Georgia State University

• Karline Feller, Georgia Perimeter College

• Ulrich Albrecht, Auburn University
• Caroline Autrey, University of West Georgia

• Terran Felter, California State University,
Bakersfield

• Robin Ayers, Western Kentucky University

• Maggie Flint, Northeast State Community College

• Robert Bass, Gardner Webb University

• Heng Fu, Thomas Nelson Community College

• Jo Battaglia, Penn State University

• Igor Fulman, Arizona State University

• Chris Bendixen, Lake Michigan College

• Abel Gage, Skagit Valley College

• Kimberly Bennekin, Georgia Perimeter College
• Allan Berele, DePaul University
• Rebecca Berg, Bowie State University
• Andrew Beyer, San Francisco State University
• Michael Boardman, Pacific University
• Bob Bradshaw, Ohlone College

• Kevin Farrell, Lyndon State College

• Gail Gonyo, Adirondack Community College
• Ivan Gotchev, Central Connecticut State
University
• Peg Greene, Florida Community College
• Michael B. Gregory, University of North Dakota
• Julio Guillen, New Jersey City University
• Mako Haruta, University of Hartford

• Ellen Brook, Cuyahoga Community College

• Judy Hayes, Lake-Sumter Community College

• David Buhl, Northern Michigan University

• Richard Hill, Idaho State University

• William L. Burgin, Gaston College

• Alan Hong, Santa Monica College

• Brenda Burns-Williams, North Carolina State
University

• Mizue Horiuchi, San Francisco State University

• Nick Bykov, San Joaquin Delta College
• Keith G. Calkins, Andrews University

• Miles Hubbard, St. Cloud State University
• Stacey Hubbard, San Francisco State University
• Brian Jue, California State University, Stanislaus

• Tom Caplinger, University of Memphis

• Dongrim Kim, Arizona State University

• Jamylle Carter, San Francisco State University
• Yu Chen, Idaho State University

• Mohammed Kazemi, University of North

Carolina, Charlotte

• Charles Conley, University of North Texas

• Curtis Kifer, San Francisco State University

• Robert Crise Jr., Crafton Hills College

• Betty Larson, South Dakota State University

• Joanne Darken, Community College of
Philadelphia

• Richard Leedy, Polk Community College

• Joyati Debnath, Winona State University

• Richard Low, San José State University

• Donna Densmore, Bossier Parish Community
College

• Jane Mays, Grand Valley State University

• Jeff Dodd, Jacksonville State University

• Scot Morrison, Western Nevada College

• Benay Don, Suffolk County Community College

• Scott Mortensen, Dixie State College

• Marcia Drost, Texas A & M University

• Susan Nelson, Georgia Perimeter College

• Douglas Dunbar, Okaloosa-Walton Community
College

• Nicholas Passell, University of Wisconsin, Eau
Claire

• Jason Edington, Mendocino College

• Vic Perera, Kent State University, Trumbull

• Mary Legner, Riverside Community College

• Eric Miranda, San Francisco State University


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xxii

Acknowledgments

• David Ray, University of Tennessee, Martin

• Janet Tarjan, Bakersfield College

• Alexander Retakh, Stony Brook University

• Chia-chi Tung, Minnesota State University

• Erika Rhett, Claflin University

• Hanson Umoh, Delaware State University

• Randy Ross, Morehead State University

• Charles Waiveris, Central Connecticut State
University

• Carol Rychly, Augusta State University
• David Santos, Community College of
Philadelphia
• Virginia Sheridan, California State University,
Bakersfield
• Barbara Shipman, University of Texas, Arlington
• Tatiana Shubin, San José State University
• Dave Sobecki, Miami University
• Jacqui Stone, University of Maryland
• Karel Stroethoff, University of Montana, Missoula

• Jeff Waller, Grossmont College
• Amy Wangsness, Fitchburg State College
• Rachel Winston, Cerro Coso Community College
• Elizabeth Wisniewski, Marymount College of
Fordham University

• Tzu-Yi Alan Yang, Columbus State Community
College
• David Zeigler, California State University,
Sacramento


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Preface to the Student

This book will help prepare you to succeed in calculus. If you master the
material in this book, you will have the knowledge, the understanding, and
the skills needed to do well in a calculus course.
To learn this material well, you will need to spend serious time reading this
book. You cannot expect to absorb mathematics the way you devour a novel.
If you read through a section of this book in less than an hour, then you are
going too fast. You should pause to ponder and internalize each definition,
often by trying to invent some examples in addition to those given in the
book. For each result stated in the book, you should seek examples to show
why each hypothesis is necessary. When steps in a calculation are left out
in the book, you need to supply the missing pieces, which will require some
writing on your part. These activities can be difficult when attempted alone;
try to work with a group of a few other students.
You will need to spend several hours per section doing the exercises and
problems. Make sure that you can do all the exercises and most of the problems, not just the ones assigned for homework. By the way, the difference
between an exercise and a problem in this book is that each exercise has a
unique correct answer that is a mathematical object such as a number or a
function. In contrast, the solutions to problems consist of explanations or
examples; thus problems have multiple correct answers.
Have fun, and best wishes in your studies!

Sheldon Axler
San Francisco State University
web page: www.axler.net

xxiii

Complete worked-out
solutions to the oddnumbered exercises
are given at the end of
each section.